## Journal of Symbolic Logic

### Computably isometric spaces

Alexander G. Melnikov

#### Abstract

We say that an uncountable metric space is computably categorical if every two computable structures on this space are equivalent up to a computable isometry. We show that Cantor space, the Urysohn space, and every separable Hilbert space are computably categorical, but the space $\mathcal{C}[0,1]$ of continuous functions on the unit interval with the supremum metric is not. We also characterize computably categorical subspaces of $\mathbb{R}^n$, and give a sufficient condition for a space to be computably categorical. Our interest is motivated by classical and recent results in computable (countable) model theory and computable analysis.

#### Article information

Source
J. Symbolic Logic, Volume 78, Issue 4 (2013), 1055-1085.

Dates
First available in Project Euclid: 5 January 2014

https://projecteuclid.org/euclid.jsl/1388953994

Digital Object Identifier
doi:10.2178/jsl.7804030

Mathematical Reviews number (MathSciNet)
MR3156512

Zentralblatt MATH identifier
1332.03008

#### Citation

Melnikov, Alexander G. Computably isometric spaces. J. Symbolic Logic 78 (2013), no. 4, 1055--1085. doi:10.2178/jsl.7804030. https://projecteuclid.org/euclid.jsl/1388953994